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functional. I. Spherical nuclei
T. Lesinski, M. Bender, K. Bennaceur, T. Duguet, J. Meyer
To cite this version:
hal-00140169, version 4 - 27 Jul 2007
T. Lesinski,1, ∗ M. Bender,2, 3, † K. Bennaceur,1, 2 T. Duguet,4 and J. Meyer1
1Universit´e de Lyon, F-69003 Lyon, France; Institut de Physique Nucl´eaire de Lyon,
CNRS/IN2P3, Universit´e Lyon 1, F-69622 Villeurbanne, France
2DSM/DAPNIA/SPhN, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France
3Universit´e Bordeaux 1; CNRS/IN2P3; Centre d’ ´Etudes Nucl´eaires de Bordeaux Gradignan,
UMR5797, Chemin du Solarium, BP120, F-33175 Gradignan, France
4National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA (Dated: April 4, 2007)
We perform a systematic study of the impact of the J2tensor term in the Skyrme energy functional
on properties of spherical nuclei. In the Skyrme energy functional, the tensor terms originate both from zero-range central and tensor forces. We build a set of 36 parameterizations which cover a wide range of the parameter space of the isoscalar and isovector tensor term coupling constants with a fit protocol very similar to that of the successful SLy parameterizations. We analyze the impact of the tensor terms on a large variety of observables in spherical mean-field calculations, such as the spin-orbit splittings and single-particle spectra of doubly-magic nuclei, the evolution of spin-orbit splittings along chains of semi-magic nuclei, mass residuals of spherical nuclei, and known anomalies of radii. The major findings of our study are (i) tensor terms should not be added perturbatively to existing parameterizations, a complete refit of the entire parameter set is imperative. (ii) The free
variation of the tensor terms does not lower the χ2 within a standard Skyrme energy functional.
(iii) For certain regions of the parameter space of their coupling constants, the tensor terms lead to instabilities of the spherical shell structure, or even the coexistence of two configurations with different spherical shell structure. (iv) The standard spin-orbit interaction does not scale properly with the principal quantum number, such that single-particle states with one or several nodes have too large spin-orbit splittings, while those of nodeless intruder levels are tentatively too small. Tensor terms with realistic coupling constants cannot cure this problem. (v) Positive values of the coupling constants of proton-neutron and like-particle tensor terms allow for a qualitative description of the evolution of spin-orbit splittings in chains of Ca, Ni and Sn isotopes. (vi) For the same values of the tensor term coupling constants, however, the overall agreement of the single-particle spectra in doubly-magic nuclei is deteriorated, which can be traced back to features of the single-particle spectra that are not related to the tensor terms. We conclude that the currently used central and spin-orbit parts of the Skyrme energy density functional are not flexible enough to allow for the presence of large tensor terms.
PACS numbers: 21.10.Dr, 21.10.Pc, 21.30.Fe, 21.60.Jz
I. INTRODUCTION
The strong nuclear spin-orbit interaction in nuclei is responsible for the observed magic numbers in heavy nu-clei [1, 2, 3, 4]. While a simple spin-orbit interaction al-lows for the qualitative description of the global features of shell structure, the available data suggest that single-particle energies evolve with neutron and proton number in a manner that cannot be related to the geometrical growth of the single-particle potential with N and Z. Many anomalies of shell structure have been identified that do not fit into simple experimental systematics, and that challenge any global model of nuclear structure.
The evolution of shell structure with N and Z as a fea-ture of self-consistent mean-field models has been known for long. To quote the pioneering study of shell structure
∗Electronic address: [email protected] †Electronic address: [email protected]
quenching” to explain the abundance pattern from the astrophysical r-process of nucleosynthesis [9, 10, 11, 12]. These two effects take place in neutron-rich nuclei. In proton-rich nuclei, the Coulomb barrier suppresses both the diffuseness of the proton density and the coupling of bound proton states to the continuum. But the Coulomb interaction itself can also modify the shell structure: for super-heavy nuclei, it begins to destabilize the nucleus as a whole. Mean-field models predict that it ampli-fies the shell oscillations of the densities for incomplete filled oscillator shells, which leads to strong variations of the density profile that feed back onto the single-particle spectra [13, 14].
Interestingly, most theoretical papers about the evolu-tion of shell structure from the last decade have specu-lated about new effects that mainly affect neutron shells in nuclei far from stability in the anticipation of the rare-isotope physics that might become accessible with the next generation of experimental facilities. The known anomalies, some of which have been known for a long time, and many more have been identified recently, con-cern also proton shells and already appear sufficiently close to stability that “exotic phenomena can be ruled out for their explanation” in most cases, to paraphrase the authors of Ref. [15]. By contrast, this suggests that there exists a mechanism that induces a strong evolution of single-particle spectra already in stable nuclei that has been overlooked for long.
There is a prominent ingredient of the nucleon-nucleon interaction that has been ignored for decades in virtu-ally all global nuclear structure models for medium and heavy nuclei, be it macroscopic-microscopic approaches or self-consistent mean-field methods. It is only very re-cently, that the systematic discrepancies between model predictions and experiment have triggered a renaissance of the tensor force in the description of finite medium-and heavy-mass nuclei.
The tensor force is a crucial and necessary ingredient of the bare nucleon-nucleon interaction [16, 17], and con-sequently is contained in all ab-initio approaches that are available for light, mainly p-shell nuclei [18, 19]. One of the first experimental signatures of the tensor force was the small, but finite quadrupole moment of the deuteron. In a boson-exchange picture of the bare nucleon-nucleon interaction, the tensor force originates from the exchange of pseudoscalar pions, which have both central and tensor couplings, see for example section 2.3 in Ref. [20] or ap-pendix 13A of Ref. [21]. In a nuclear many-body system, the bare tensor force induces a strong correlation between the spatial and spin orientations in the two-body density matrix. For two nucleons with parallel spins, the ten-sor force energetically favors the configuration where the distance vector is aligned with the spins, while for anti-parallel spins the tensor force prefers when the distance vector is perpendicular to the spins, see the discussion of Fig. 13 in Ref. [22] and of Fig. 3 in Ref. [23]. The authors of these papers also demonstrate very nicely the well-known fact [24, 25] that in an approach that starts
from the bare nucleon-nucleon interaction, nuclei are not bound without taking into account the two-body corre-lations induced by the tensor force.
The role of the tensor force, however, manifests itself differently in self-consistent mean-field models, otherwise called energy density functional (EDF) methods, the tool of choice for medium and heavy nuclei. The latter meth-ods use an independent-particle state as a reference state to express the energy of the correlated nuclear ground state. Thus, correlations are not explicitly present in the higher-order density matrices of the reference state, but rather included under the form of a more elaborate func-tional of the (local and nearly local parts of the) one-body density matrix of that reference state. In such a scheme, most of the effect of the bare tensor force on the binding energy is integrated out through the renormalization of the coupling constants associated with a central effective vertex, in a similar fashion as the tensor part of the bare interaction is renormalized into the central one when go-ing from the bare nucleon-nucleon force to a Brueckner G matrix. The tensor terms of the EDF relate to a residual tensor vertex, that gives nothing but a correction to the spin-orbit splittings, which for light p-shell nuclei might be of the same order as the contribution from the genuine spin-orbit force. The interplay of spin-orbit and tensor forces in the mean field of medium and heavy nuclei was explored in Refs. [26, 27, 28], where the particular role of spin-unsaturated shells was pointed out.
There are two widely used effective interactions for non-relativistic self-consistent mean-field models [29], the zero-range non-local Skyrme interaction [30, 31, 32, 33] on the one hand and the finite-range Gogny force [34, 35] on the other hand.
In fact, the effective zero-range non-local interaction proposed by Skyrme in 1956 [30, 31, 32, 33] already con-tained a zero-range tensor force. The first applications of Skyrme’s interaction in self-consistent mean-field models that became available around 1970, however, neglected the tensor force, and the simplified effective Skyrme in-teraction used in the seminal paper by Vautherin and Brink [36] soon became the standard Skyrme interaction that was used in most applications ever since. Until very recently, there was only very little exploratory work on Skyrme’s tensor force. In their early study, Stancu, Brink and Flocard [37], who added the tensor force perturba-tively to the SIII parameterization, pointed out that some spin-orbit splittings in magic nuclei can be improved with a tensor force. A complete fit including the terms from the tensor force that contribute in spherical nuclei was attempted by Tondeur [38], with the relevant coupling constants of the spin-orbit and tensor terms adjusted to selected spin-orbit splittings in16O,48Ca and208Pb. An-other complete fit of a generalized Skyrme interaction in-cluding a tensor force was performed by Liu et al. [39], but the authors did not investigate the effect of the ten-sor force in detail, nor was the resulting parameterization ever used in the literature thereafter.
eval-uation of matrix elements of a finite-range force of Gaus-sian shape in an harmonic oscillator basis contains the ex-pressions for a finite-range tensor force, which, however, was omitted in the parameterizations of Gogny’s force
adjusted by the Bruy`eres-le-Chˆatel group [35]. It were
Onishi and Negele [40] who first published an effective interaction that combined a Gaussian two-body central force, a finite-range tensor force with a zero-range spin-orbit force and a zero-range non-local three-body force, which, however, also fell into oblivion.
The role of the tensor force is slightly different in Skyrme and Gogny interactions. In the Gogny force, the contributions from the central and tensor parts re-main explicitly distinct, although, of course, this does not prevent a certain entanglement of their physical ef-fects. In the context of Skyrme’s functional, however, the contribution of a zero-range tensor force to the spherical mean-field state of an even-even nucleus has exactly the same form as a particular exchange term from the non-local part of the central Skyrme force. When looking at spherical nuclei only, adding Skyrme’s tensor force simply allows one to decouple a term that is already provided by the central force. This indeed makes the effective-interaction-restricted functional more flexible, as the ad-ditional degrees of freedom from the tensor force remove an interdependence between the effective mass, the sur-face terms and the “tensor terms”. However, one must always keep in mind that both the central and tensor
part of the effective vertex contribute to the so-called J2
t
“tensor” terms of the functional.1
In the context of relativistic mean-field models, the equivalent of the non-relativistic tensor force appears as the exchange term of effective fields with the quan-tum numbers of the pion, which by construction do not appear in the standard relativistic Hartree models. Only relativistic Hartree-Fock models contain this tensor force, with the first predictive parameterizations becom-ing available just recently [42].
We also mention that there is a large body of work on the tensor force in the interacting shell model, see Ref. [43] for a review, that concentrates on a completely different aspect of the tensor force, namely its unique contribution to excitations with unnatural parity.
The recent interest in the effect of the tensor force in the context of self-consistent mean field models was trig-gered by the observed evolution of single-particle levels of one nucleon species in dependence of the number of the other nucleon species. Otsuka et al. [44] proposed
1 As we will outline below, and as was already pointed out in
Ref. [5], this argument does not hold for deformed even-even nu-clei or any situation where intrinsic time-reversal is broken, for example odd nuclei or dynamics. There, the tensor and non-local central parts of the effective Skyrme interaction give contribu-tions to the mean-fields and the binding energy with different analytical expressions. This will be discussed in a companion article [41].
that at least part of the effect is caused by the proton-neutron tensor force from pion exchange. Many groups attempt now to explain known, but so far unresolved, anomalies of shell structure in terms of a tensor force. A particularly popular playground is the relative shift of the proton 1g7/2and 1h11/2levels in tin isotopes, which is interpreted as the reduction of the spin-orbit splittings of both levels with their respective partners with increasing neutron number [45].
Otsuka et al. [46] added a Gaussian tensor force, ad-justed on the long-range part of a one-pion+ρ exchange potential, to a standard Gogny force. After a consis-tent readjustment of the parameters of its central and spin-orbit parts, they were able to explain coherently the anomalous relative evolution of some single-particle levels without, however, being able to describe their absolute distance in energy. Dobaczewski [47] has pointed out that a perturbatively added tensor interaction with suitably chosen coupling constants in the Skyrme energy density functional does not only modify the evolution of shell structure, but does also improve the description of nu-clear masses around magic nuclei. Brown et al. [48] have fitted a Skyrme interaction with added zero-range tensor force with emphasis on the reproduction of single-particle spectra. While the authors appreciate the qualitatively correctly described evolution of relative level distances, they point out that the combination of zero-range spin-orbit and tensor forces does not and can not correctly describe the ℓ-dependence of spin-orbit splittings. Col`o
et al. [49], and Brink et al. [50] have added Skyrme’s
tensor force perturbatively to the existing standard pa-rameterization SLy5 [51, 52], and to the SIII [5] one, respectively. They have investigated some single-particle
energy differences: the 1h11/2and 1g7/2proton states in
tin isotopes as well as 1i13/2and 1h9/2 neutron states in
N = 82 isotones and propose similar parameters as in Ref. [48]. The effect of the tensor force on the centroid of the GT giant resonance is also estimated by Col`o et al. using a sum-rule approach and found to be substantial. Long et al. [53], demonstrate that the tensor force that emerges naturally in relativistic Hartree-Fock also
im-proves the relative shifts of the proton 1g7/2 and 1h11/2
levels in tin isotopes.
in-vestigate the effect of the tensor terms on a multitude of observables in nuclei though a set of Skyrme interac-tions with systematically varied coupling constants of the tensor terms.
The present study was motivated by the finding that the performance of the existing Skyrme-type effective in-teractions for masses and spectroscopic properties is lim-ited by systematic deficiencies of the single-particle spec-tra [54, 55, 56, 57] that seem to be impossible to remove within the standard Skyrme interaction. The details of single-particle spectra were so far somewhat outside the focus of self-consistent mean-field methods, on the one hand as they do not correspond directly to empirical single-particle energies (we will come back to that be-low), and on the other hand because many of the ob-servables that are usually calculated with self-consistent mean-field methods are not very sensitive to the exact placement of single-particle levels. By contrast, there is an enormous body of work that examines the infi-nite and semi-infiinfi-nite nuclear matter properties of the effective interactions that are the analog of liquid-drop and droplet parameters in great detail. The reason is, of course, that the global trends over the whole chart of nuclei have to be understood before one can look into details. The last few years have seen an increasing de-mand on predictive power. Moreover, beyond-mean-field approaches of the projected generator coordinate method (GCM), or Bohr-Hamiltonian type, have become widely used tools to analyze and predict spectroscopic properties in medium and heavy nuclei, employing either Gogny or Skyrme interactions. The underlying single-particle tra thus now deserve more attention, as many of the spec-troscopic properties of interest turn out to be extremely sensitive to even subtle details of the single-particle spec-tra. As the tensor force is the most obvious missing piece in all standard mean-field interactions, it is the natural starting point for the systematic investigation of possi-ble generalizations with the ultimate goal to improve the predictive power of the interactions for spectroscopy.
In the present paper, we will outline the formalism of a Skyrme interaction with added tensor force, describe the fit of the parameterizations, analyze the role of the tensor terms for single-particle spectra, masses and radii of spherical even-even nuclei. A second paper [41] studies the surface and deformation properties of these Skyrme interactions for even-even nuclei, and future work will ex-amine the stability of nuclear matter and the role of the time-odd terms from the tensor force in odd and rotating nuclei. Only deformed nuclei and, in particular, observ-ables sensitive to the time-odd contributions, will pos-sibly allow to distinguish clearly between the non-local central and tensor parts of the Skyrme force.
II. THE SKYRME INTERACTION WITH
TENSOR TERMS
A. The energy density functional
The usual ansatz for the Skyrme effective interac-tion [51, 52] leads to an energy density funcinterac-tional which can be written as the sum of a kinetic term, the Skyrme potential energy functional that models the effective strong interaction in the particle-hole channel, a pairing energy functional corresponding to a density-dependent contact pairing interaction, the Coulomb energy func-tional (calculated using the Slater approximation [58]) and correction terms to approximately remove the ex-citation energy from spurious motion caused by broken symmetries
E = Ekin+ ESkyrme+ Epairing+ ECoulomb+ Ecorr. (1)
B. The Skyrme energy density functional
Throughout this work, we will use an effective Skyrme energy functional that corresponds to an antisymme-trized density-dependent two-body vertex in the particle-hole channel of the strong interaction, that can be decom-posed into a central, spin-orbit and tensor contribution
vSkyrme= vc+ vt+ vLS. (2)
Other choices for the writing of the Skyrme energy func-tional are possible and have been made in the literature, which might affect the form of the effective interaction, its interpretation and the results obtained from it. We will come back to that in section II D below.
The Skyrme energy density functional is a functional of local densities and currents
ESkyrme=
Z
d3r HSkyrme(r) , (3)
which has many technical advantages compared to finite-range forces such as the Gogny force. All exchange terms have the same structure as the direct terms, which greatly reduces the number of necessary integrations during a calculation.
1. Local densities and currents
of protons and neutrons in coordinate space [60] ρq(rσ, r′σ′) = hˆar†′σ′qˆarσqi = 12ρq(r, r′)δσσ′ +12sq(r, r′) · hσ′| ˆσ|σi (4) where ρq(r, r′) = X σ ρq(rσ, r′σ) sq(r, r′) = X σσ′ ρq(rσ, r′σ′) hσ′| ˆσ|σi . (5)
The Skyrme energy functional up to second order in derivatives that we will introduce below can be expressed in terms of seven local densities and currents [59] that are defined as ρq(r) = ρq(r, r′) r=r′ sq(r) = sq(r, r′) r=r′ τq(r) = ∇ · ∇′ρq(r, r′) r=r′ Tq,µ(r) = ∇ · ∇′sq,µ(r, r′) r=r′ jq(r) = −2i(∇ − ∇′) ρq(r, r′) r=r′ Jq,µν(r) = −2i(∇µ− ∇′µ) sq,ν(r, r′) r=r′ Fq,µ(r) = 12 z X ν=x ∇µ∇′ν+ ∇′µ∇ν sq,ν(r, r′) r=r′(6)
which are the density ρq(r), the kinetic density τq(r),
the current (vector) density jq(r), the spin
(pseudovec-tor) density sq(r), the spin kinetic (pseudovector) density
Tq(r), the spin-current (pseudotensor) density Jq,µν(r),
and the tensor-kinetic (pseudovector) density Fq(r).
ρq(r), τq(r) and Jq,µν(r) are time-even, while sq(r), Tq(r), jq(r) and Fq(r) are time-odd. For a detailed dis-cussion of their symmetries see Ref. [60]. There are other local densities up to second order in derivatives that can be constructed, but when constructing an energy func-tional they either cannot be combined with others to terms with proper symmetries or they lead to terms that are not independent from the others [61].
The cartesian spin-current pseudotensor density Jµν
can be decomposed into pseudoscalar, (anti-symmetric) vector and (symmetric) traceless pseudotensor parts, all of which have well-defined transformation properties un-der rotations Jµν(r) = 13δµνJ(0)(r) +12 z X κ=x ǫµνκJκ(1)(r) + Jµν(2)(r) , (7)
where δµν is the Kronecker symbol and ǫµνκ the
Levi-Civita tensor. The pseudoscalar, vector and pseudoten-sor parts expressed in terms of the cartesian tenpseudoten-sor are
given by J(0)(r) = z X µ=x Jµµ(r) , (8) Jκ(1)(r) = z X µ,ν=x ǫκµνJµν(r) , Jµν(2)(r) = 12[Jµν(r) + Jνµ(r)] − 1 3δµν z X κ=x Jκκ(r) .
The vector spin current density J(1)(r) ≡ J(r) is often
called spin-orbit current, as it enters the spin-orbit energy density. 2
For the formal discussion of the physical content of the Skyrme energy functional it is of advantage to recouple the proton and neutron densities to isoscalar and isovec-tor densities, for example
ρ0(r) = ρn(r) + ρp(r) ,
ρ1(r) = ρn(r) − ρp(r) (9)
and similar for all others. As we assume pure proton
and neutron states, only the Tz = 0 component of the
isovector density is non-zero, which we exploit to drop the index Tz from the isovector densities ρ1Tz(r) etc.
2. Skyrmes’s central force
We will use the standard density-dependent central Skyrme force vc(R, r) = t0(1 + x0Pˆσ) δ(r) + 16t3(1 + x3Pˆσ) ρα(R) δ(r) + 1 2t1(1 + x1Pˆσ) ˆ k′2δ(r) + δ(r) ˆk2 + t2 (1 + x2Pˆσ) ˆk′· δ(r) ˆk (10)
where we use the shorthand notation r = r1− r2,
R = 1
2(r1+ r2) , (11)
while ˆkis the usual operator for relative momenta
ˆ
k= −2i(∇1− ∇2) (12)
and ˆk′its complex conjugated acting on the left. Finally, ˆ
Pσis the spin exchange operator that controls the relative
strength of the S = 0 and S = 1 channels for a given term in the two-body interaction
ˆ
Pσ= 12(1 + ˆσ1· ˆσ2) . (13)
2Some authors call J(r) spin density, which is ambiguous and
As said above, we restrict ourselves to a parameterization of the Skyrme energy functional as obtained from the average value of an effective two-body vertex in the ref-erence Slater determinant. We decompose the isoscalar and isovector parts of the resulting energy density func-tional Hcinto a part Hc,even
t that is composed entirely of
time-even densities and currents, and a part Hc,oddt that
contains terms which are bilinear in time-odd densities and currents and vanishes in intrinsically time-reversal invariant systems Hc(r) = X t=0,1 Hc,event (r) + H c,odd t (r) . (14)
Both Hc,event and H
c,odd
t are of course constructed such
that they are time-even; they are given by [59, 62]
Htc,even = A ρ t[ρ0] ρ2t+ A ∆ρ t ρt∆ρt+ Aτtρtτt −ATt z X µ,ν=x Jt,µνJt,µν, Hc,oddt = Ast[ρ0] s2t− Aτtj2t +A∆st st· ∆st+ ATt st· Tt, (15) where Aρt[ρ0] and Ast[ρ0] are density dependent coupling constants that depend on the total (isoscalar) density. The detailed relations between the coupling constants of the functional and the central Skyrme force are given in appendix A. The notation reflects that two pairs of terms in Hc,event and H
c,odd
t are connected by the
require-ment of local gauge invariance of the Skyrme energy func-tional [63].
3. A zero-range spin-orbit force
The spin-orbit force used with most standard Skyrme interactions
vLS(r) = iW
0( ˆσ1+ ˆσ2) · ˆk′× δ(r) ˆk (16) is a special case of the one proposed by Bell and Skyrme [32, 33]. Again, the corresponding energy
func-tional [59, 62] can be separated into a time-even and a time-odd term HLS(r) = X t=0,1 HLS,even t (r) + HLS,oddt (r) (17) where HtLS,even = A∇·Jt ρt∇ · Jt HLS,oddt = A∇·Jt st· ∇ × jt (18)
which share the same coupling constant as again both terms are linked by the local gauge invariance of the
en-ergy functional. The relation between the A∇·J
t and the
one coupling constant of the two-body spin-orbit force
W0is given in appendix A.
4. Skyrme’s tensor force
By convention, the tensor operator in the tensor force is constructed using the unit vectors in the direction of the relative coordinate er= r/|r| and subtracting ˆσ1· ˆσ2
ˆ
S12= 3( ˆσ1· er)( ˆσ2· er) − ˆσ1· ˆσ2, (19) such that its mean value vanishes for a relative S state, which decouples the central and tensor channels of the
interaction. The operator ˆS12 commutes with the total
spin [ ˆS12, ˆS2] = 0, therefore it does not mix partial waves with different spin, i.e. spin singlet and spin triplet states. In particular, it does not act in spin singlet states at all, as ˆS12PˆS=0 = 0 (see section 13.6 of Ref. [21]). As a consequence, there is no point in multiplying a tensor force with an exchange operator (1+xtPˆσ) as done for the central force, as this will only lead to an overall rescaling of its strength.
The derivation of the general energy functional from a zero-range two-body tensor force is discussed in detail in Refs. [59, 64]. We repeat here the details relevant for our discussion, starting from the two zero-range tensor forces proposed by Skyrme [30, 31] vt(r) = 1 2te n 3 (σ1· k′) (σ2· k′) − (σ1· σ2) k′2 δ(r) + δ(r) 3 (σ1· k) (σ2· k) − (σ1· σ2) k2 io +to h 3 (σ1· k′) δ(r) (σ2· k) − (σ1· σ2) k′· δ(r) k i (20)
where r, ˆkand ˆk′are defined as above, Eqs. (11) and (12). The corresponding energy density functional can again be
decomposed in a time-even and a time-odd part
with [59] Ht,event = −BTt z X µ,ν=x Jt,µνJt,µν −12BtF Xz µ=x Jt,µµ 2 −1 2B F t z X µ,ν=x Jt,µνJt,νµ Htt,odd = BtTst· Tt+ BtFst· Ft +B∆st st· ∆st+ Bt∇s(∇ · st)2, (22) where we already used the local gauge invariance of the energy functional [59] for the expressions of the coupling constants. The actual expressions for the coupling con-stants expressed in terms of the two coupling concon-stants
te and to of the tensor forces are given in appendix A.
The “even” term proportional to te in the two-body
tensor force (20) mixes relative S and D waves, while
the “odd” term proportional to to mixes relative P and
F waves. Thus, due to the fact that both act in spin-triplet states only, antisymmetrization implies that the former acts in isospin-singlet states (and hence con-tributes to the neutron-proton interaction only) and the latter in isospin-triplet states (contributing both to the like-particle and neutron-proton interactions). The
cen-tral and spin-orbit interactions as we use them, however, do not contain D or F wave interactions. From this point of view, one might suspect a mismatch when combining the various interaction terms. From the point of view of the energy functional (22), however, all contributions from the zero-range tensor force are of the same second order in derivatives as the contributions from the non-local part of the central Skyrme force (15) and from the spin-orbit force (18).
In the time-even part of the energy functional Ht,event , there appear three different combinations of the carte-sian components of the spin current tensor. The term
proportional to BT
t contains the symmetric combination
JµνJµν as it already appeared in the energy functional
from the central Skyrme interaction (15), while the term
proportional to BF
t contains two different terms, namely
the antisymmetric combination JµνJνµand the square of
the trace of Jνµ.
5. Combining central and tensor interactions
The Skyrme energy functional representing central, tensor, and spin-orbit interactions is given by
ESkyrme = Ec+ ELS+ Et = Z d3r X t=0,1 Ctρ[ρ0] ρ2t+ Cts[ρ0] s2t+ C ∆ρ t ρt∆ρt+ Ct∇s(∇ · st)2+ Ct∆sst· ∆st+ Ctτ(ρtτt− j2t) +CtT st· Tt− z X µ,ν=x Jt,µνJt,µν + CtF h st· Ft−12 Xz µ=x Jt,µµ 2 −1 2 z X µ,ν=x Jt,µνJt,νµ i +C∇·J t (ρt∇ · Jt+ st· ∇ × jt) . (23)
This functional contains all possible bilinear terms up to second order in the derivatives that can be constructed from local densities and that are invariant under spatial and time inversion, rotations, and local gauge transfor-mations [59].
Some of the coupling constants are completely defined
by the standard central Skyrme force, i.e. Ctρ = A
ρ t, Cs t = Ast, Ctτ = Aτt, and C ∆ρ t = A ∆ρ t , two by the
spin-orbit force, Ct∇J = A∇Jt , others by the tensor force, CF
t = BFt and Ct∇s = Bt∇s, while some are the sum of
coupling constants from both central and tensor forces, CT
t = ATt + BtT, and Ct∆s= A∆st + Bt∆s.
The three terms bilinear in Jµν can be recoupled into
terms bilinear in its pseudoscalar, vector, and
pseudoten-sor components J(0), J(1), and J(2), Eq. (8), which is
prefered by some authors [59]
After combining (23) with the kinetic, Coulomb, pairing and other contributions from (1), the mean-field equa-tions are obtained by standard functional derivative tech-niques from the total energy functional [29, 59].
The complete Skyrme energy functional (23) has quite complicated a structure, and in the most general case leads to seven distinct mean fields in the single-particle Hamiltonian [59]. As already mentioned, we want to di-vide the examination of those terms that contain two derivatives and two Pauli matrices in the complete func-tional, i.e. those terms from the central Skyrme force that are often neglected and all the terms from the ten-sor Skyrme force, into three distinct steps: First, in the present paper, we enforce spherical symmetry which re-moves all time-odd densities and all but one out of the
nine components of the spin current tensor Jµν as will
be outlined in the following section. A subsequent pa-per [41] will discuss deformed even-even nuclei where the
complete spin current tensor Jµν is present, and future
work will address the time-odd part of the energy func-tional (23).
C. The Skyrme energy functional in spherical
symmetry
For the rest of this paper, we will concentrate on spher-ical nuclei, enforcing spherspher-ical symmetry of the N -body wave functions. As a consequence, the canonical
single-particle wave functions Ψi [65] can be labeled by ji, ℓi
and mi. The index ni labels the different states with
same ji and ℓi. The functions Ψi separates into a radial
part ψ and an angular and spin part, represented by a
tensor spherical harmonic Ωjℓm
Ψnjℓm(r) = 1rψnjℓ(r) Ωjℓm(θ, φ) . (26)
Spherical symmetry also enforces that all magnetic
sub-states of Ψnjℓm have the same occupation probability
v2
njℓm ≡ vnjℓ2 for all −j ≤ m ≤ j. For a static spherical
state, all time-odd densities are zero sq(r) = Tq(r) =
jq(r) = Fq(r) = 0, as are the corresponding mean fields
in the single-particle Hamiltonian.
Enforcing spherical symmetry also greatly simplifies the spin-current tensor, both the pseudoscalar and
pseu-dotensor parts of Jµν vanish. From the vector spin-orbit
current, only the radial component is non-zero, which is given by [36] Jq(r) = 1 4πr3 X n,j,ℓ (2j + 1) vnjℓ2 ×hj(j + 1) − ℓ(ℓ + 1) −34iψnjℓ2 (r) (27) so that there is only one out of the nine components of the spin-current tensor density that contributes in spherical nuclei. Unlike the total density ρ and the kinetic den-sity τ , that are bulk properties of the nucleus and grow with the size of the nucleus, the spin-orbit current is a
shell effect that shows strong fluctuations. Assume the two shells with same n and ℓ which are split by the spin-orbit interaction, one coupled with the spin to j = ℓ +12,
the other to j = ℓ − 12. It is easy to verify that their
contributions to Jq(r) are equal but of opposite signs
such that they cancel when (i) both shells are completely filled and (ii) their radial wave functions are identical
ψn,ℓ+1/2,ℓ = ψn,ℓ−1/2,ℓ. Although the latter condition is
never exactly fulfilled, this demonstrates that the spin-orbit current is not a bulk property, but a shell effect that strongly fluctuates with N and Z. It nearly van-ishes in so-called saturated nuclei, where all spin-orbit partners are either completely occupied or empty, and it might be quite large when only the j = ℓ+1/2 level out of one or even several pairs of spin-orbit partners is filled.
Altogether, the Skyrme part of the energy density func-tional in spherical nuclei is reduced to
HSkyrme = X t=0,1 n Ctρ[ρ0] ρ2t+ C ∆ρ t ρt∆ρt+ Ctτρtτt +12CtJJ2t+ Ct∇·Jρt∇ · Jt o , (28)
where we have introduced an effective coupling constant CJ
t of the J2t tensor terms at sphericity, such that the
corresponding contribution to the energy functional is given by Ht= X t=0,1 1 2C J t J2t = X t=0,1 −12CtT +14C F t J2t. (29)
The effective coupling constants can be separated back into contributions from the non-local central and tensor forces
CJ
t = AJt + BtJ (30)
which are given by AJ0 = 18t1 1 2− x1 − 1 8t2 1 2+ x2 AJ1 = 161 t1− 1 16t2 B0J = 165 (te+ 3to) = 5 48(T + 3U ) B1J = 165 (to− te) = 5 48(U − T ) , (31)
where we also give the expressions using the notation
T = 3teand U = 3to employed in [37, 49, 64].
The proton-neutron coupling constants α = αC+ αT and
β = βC+ βT can again be separated into contributions
from central and tensor forces
αC = 18(t1− t2) −18(t1x1+ t2x2) , βC = −18(t1x1+ t2x2) ,
αT = 54to= 125 U ,
βT = 58(te+ to) = 245 (T + U ) . (34)
As could be expected, the isospin-singlet tensor force contributes only to the proton-neutron term, while the isospin-triplet tensor force contributes to both.
The spin-orbit potential of the neutrons is given by
Wn(r) = δE δJn(r) · er = W0 2 2∇ρn+ ∇ρp) + α Jn+ β Jp. (35)
The expression for the protons is obtained exchanging the indices for protons and neutrons. In spherical sym-metry, the tensor force gives a contribution to the spin-orbit potential, but does not alter the structure of the spin-orbit terms in the single-particle Hamiltonian as such. This will be different in the case of deformed mean fields [41, 59].
The dependence of the spin-orbit potential Wq(r) on
the spin-orbit current Jq(r) through the tensor terms is
the source of a potential instability. When the spin-orbit splitting becomes larger than the splitting of the cen-troids of single-particle states with different orbital angu-lar momentum ℓ, the reordering of levels might increase the number of spin-unsaturated levels, which increases
the spin-orbit current Jnand feeds back on the spin-orbit
potential by increasing it even further, which ultimately leads to an unphysical shell structure. An example will be given in appendix B.
D. A brief history of tensor terms in the central
Skyrme energy functional
For the interpretation of the parameterizations we will describe below it is important to point out that within our choice of the effective Skyrme interaction as an an-tisymmetrized vertex the two coupling constants of the contribution from the central force to HT, Eq. (29), either
represented through AJ
0, AJ1 or through αC, βC, are not independent from the coupling constants Aτ0, Aτ1, A∆ρ0 ,
and A∆ρ1 , that appear in Eq. (28). Through the
expres-sions given in appendix A, all six of them are determined by the four coupling constants t1, x1, t2, and x2from the central Skyrme force, Eq. (10). As a consequence, a ten-sor force is absolutely necessary to decouple the values of
the CJ
t from those of the Ctτ and C
∆ρ
t , which determine
the isoscalar and isovector effective masses and give the dominant contribution to the surface and surface asym-metry coefficients, respectively.
This interpretation of the Skyrme interaction is, how-ever, far from being common practice and a source of confusion and potential inconsistencies in the literature. Many authors have used parameterizations of the central and spin-orbit Skyrme energy functional with coupling constants that in one way or the other do not exactly correspond to the functional obtained from Eqns. (10) and (16), which, depending on the point of view, can be seen as an approximation to or a generalization of the original Skyrme interaction. As the most popular mod-ification concerns the tensor terms, a few comments on the subject are in order. Again, the practice goes back to the seminal paper by Vautherin and Brink [36], who state that “the contribution of this term to [the sporbit potential] is quite small. Since it is difficult to in-clude such a term in the case of deformed nuclei, it has been neglected”. This choice was further motivated by the interpretation of the effective Skyrme interaction as a density-matrix expansion (DME) [25, 66, 67, 68]. All early parameterizations as SI and SII [36], SIII-SVI [5],
SkM [69] and SkM∗ [70] followed this example and did
not contain the J2 terms. Beiner et al. [5] weakened
the case for J2 terms further by pointing out that they
might lead to unphysical single-particle spectra. During the 1980s and later, however, it became more popular to include them, for example in SkP [65], the
parame-terizations T1-T9 by Tondeur et al. [71], Eσ and Zσ by
Friedrich and Reinhard [72]. Some of the recent param-eterizations come in pairs, where variants without and
with J2 terms are fitted within the same fit protocol, for
example (SLy4, SLy5) and (SLy6, SLy7) in Ref. [52], or (SkO, SkO’) in Ref. [73].
Interestingly, all but one parameterization of the cen-tral Skyrme interaction found in the literature set the
coupling constants of the J2terms either to their Skyrme
force value (A1) or strictly to zero. The exception is Ref. [38] by Tondeur, where an independent fit of the
cou-pling constants of the J2 terms was attempted, making
explicit reference to a DME interpretation of the energy functional.
Setting the coupling constants of a term to zero when one does not know how to adjust its parameters is of course an acceptable practise when permitted by the cho-sen framework. For Skyrme interactions fitted without
the J2 terms, the situation becomes confusing when one
looks at deformed nuclei and any situation that breaks time-reversal invariance. First of all, Galilean invariance of the energy functional dictates that the coupling con-stant of the s · T terms is also set to zero, as already indicated by the presentation of the energy functional in Eq. (23). Second, using a DME interpretation of the Skyrme energy functional in one place, but the interre-lations from the two-body Skyrme force in all others is
not entirely satisfactory. Many authors who drop the J2
terms rarely show scruples to keep most of the time-odd terms in the Skyrme energy functional (23) with coupling
constants As
t and A∆st from (A1), although they are not
employ-ing properties of even-even nuclei and spin-saturated nu-clear matter. For a list of exceptions see Sect. II.A.2.d of Ref. [29]. An alternative is to set up a hierarchy of terms, as it was attempted by Bonche, Flocard and Heenen in
their mean-field and beyond codes, which set A∆s
t = 0 in
addition to the coupling constant of the J2terms, as all
three terms have in common that they couple two Pauli matrices with two derivatives in different manners, see the footnote on page 129 of [74].
There are also inconsistent applications of
parameter-izations without J2− s · T terms to be found in the
lit-erature. For example, almost all applications of Skyrme
interactions to the Landau parameters gℓand gℓ′ and the
properties of polarized nuclear matter, include the con-tribution from the s · T terms, although it should be
dropped for parameterizations fitted without J2 terms.
Similarly, most RPA and QRPA codes include them for simplicity, see the discussion in Refs. [75, 76, 77].
As it is relevant for the subject of the present paper, we also mention another generalization of the Skyrme in-teraction that invokes the interpretation of the Skyrme energy functional in a DME framework. The spin-orbit force (16) fixes the isospin mix of the corresponding terms in the Skyrme energy functional (23) such that
A∇J
0 = 3A∇J1 (A2). There are a few parameterizations
as MSkA [78], SkI3 and SkI4 [79], SkO and SkO’ [73] and SLy10 [52] that liberate the isospin degree of freedom in the spin-orbit functional. A DME interpretation of the energy functional is mandatory for this generalization. It is motivated by the better performance of standard rela-tivistic mean-field models for the kink of the charge radii in Pb isotopes. Note that the standard RMF models are effective Hartree theories without exchange terms, and that the standard Lagrangians have very limited isovec-tor degrees of freedom [29], both of which supress a strong isospin dependence of the sporbit interaction. It is in-teresting to note that the existing fits of Skyrme energy functionals with generalized spin-orbit interaction do not improve spin-orbit splittings [14].
III. THE FITS
A. General remarks
In order to study the effect of the J2 terms, we have
built a set of 36 effective interactions that systematically
cover the region of coupling constants CJ
0 and C1J that give a reasonable description of finite nuclei in connec-tion with the standard central and spin-orbit Skyrme forces. At variance with the perturbative approach used in Refs. [37, 49], each of these parameterizations has been fitted separately, following a procedure nearly identical to that used for the construction of the SLy parameteriza-tions [51, 52], so that we can keep the connection between the new fits with parameterizations that have been ap-plied to a large variety of observables and phenomena. The Saclay-Lyon fit protocol focuses on the simultaneous
reproduction of nuclear bulk properties such as binding energies and radii of finite nuclei and the empirical char-acteristics of infinite nuclear matter (i.e. symmetric and pure neutron matter). The latter establishes an impor-tant, though highly idealized, limiting case as it permits to confront the energy functional with calculations from first principles using the bare nucleon-nucleon force [80].
The region of effective coupling constants (CJ
0, C1J) of
the J2 terms acting in spherical nuclei, as defined in
Eq. (28), that we will explore, is shown in Fig. 1. The parameterizations are labeled TIJ, where indices I and J refer to the proton-neutron (β) and like-particle (α) coupling constants in Eq. (32) such that
α = 60 (J − 2) MeV fm5,
β = 60 (I − 2) MeV fm5. (36)
The corresponding values of CJ
t can be obtained through
Eq. (33) or from Fig. 1. On the one hand, we cover the positions of the most popular existing Skyrme
in-teractions that take the J2 terms from the central force
into account, which are SLy5 [52], SkP [65], Zσ [72],
T6 [71], SkO’ [73] and BSk9 [81]. On the other hand, among recent parameterizations including a tensor term, i.e. Skxta [48], Skxtb [48, 82] as well as those published
by Col`o et al. [49] and Brink and Stancu [50], most fall
in a region of negative CJ
1 and vanishing C0J, that is to the lower left of Fig. 1. Parameterizations of this region, which also includes a part of the triangle advocated in the perturbative study of Stancu et al. [37], gave unsat-isfactory results for many observables. Moreover, when attempting to fit parameterizations with large negative coupling constants, we sometimes obtained unrealistic single-particle spectra or even ran into the instabilities al-ready mentioned and outlined in appendix B. Parameter-izations further to the lower and upper right also have un-realistic deformations properties. The contribution from
the J2 terms vanishes for T22, which will serve as the
reference point. For the parameterizations T2J, only the
proton-proton and neutron-neutron terms in Htare
non-zero (β = 0), while for the parameterizations TI2, only
the proton-neutron term in Htcontributes (α = 0). Note
that the earlier parameterizations T6 and Zσhave a pure
like-particle J2 terms as a consequence of the constraint
x1 = x2 = 0 employed for both (and most other early
parameterizations of Skyrme’s interaction).
B. The fit protocol and procedure
The list of observables used to construct the cost
function χ2 minimized during the fit (see Eq. (4.1) in
Ref. [51]) reads as follows: binding energies and charge radii of40Ca,48Ca,56Ni,90Zr,132Sn and208Pb; the bind-ing energy of100Sn; the spin-orbit splitting of the neutron
3p state in208Pb; the empirical energy per particle and
-150 -120 -90 -60 -30 0 30 60 90 120 150 -60 -30 0 30 60 90 120 150 180 210 240 270 C J [MeV fm1 5 ] CJ0 [MeV fm5] T11 T12 T13 T14 T15 T16 T21 T22 T23 T24 T25 T26 T31 T32 T33 T34 T35 T36 T41 T42 T43 T44 T45 T46 T51 T52 T53 T54 T55 T56 T61 T62 T63 T64 T65 T66 SLy4 SLy5 SkP SkO’ BSk9 T6 Zσ Skxta Skxtb Colo Brink FIG. 1: Values of CJ
0 and C1J for our set of
parameteriza-tions (circles). Diagonal lines indicate α = C0J+C1J= 0 (pure
neutron-proton coupling) and β = CJ
0 −C1J = 0 (pure
like-particle coupling). Values for classical parameter sets are also indicated (dots), with SLy4 representing all parameterizations
for which J2 terms have been omitted in the fit. Recent
pa-rameterizations with tensor terms are indicated by squares.
Furthermore, some properties of infinite nuclear mat-ter are constrained through analytic relations between coupling constants in the same manner as they were in
Refs. [51, 52]: the incompressibility modulus K∞ is kept
at 230 MeV, while the volume symmetry energy coeffi-cient aτis set to 32 MeV. The isovector effective mass, ex-pressed through the Thomas-Reiche-Kuhn sum rule
en-hancement factor κv, is taken such that κv= 0.25.
When using a single density-dependent term in the central Skyrme force (10), the isoscalar effective mass m∗
0cannot be chosen independently from the
incompress-ibility modulus for a given exponent α of ρ0. We
fol-low here the prescription used for the SLy parameteriza-tions [51, 52] and use α = 1/6, which leads to an isoscalar effective mass close to 0.7 in units of the bare nucleon mass for all TIJ parameterizations. This value allows for a correct description of dynamical properties, as for ex-ample the energy of the giant quadrupole resonance [83]. Using such a protocol we cannot reproduce the isovec-tor effective mass consistent with recent ab-initio predic-tions [84]. Regarding the present exploratory study of the tensor terms this is not a critical limitation, in particular as the influence of this quantity on static properties of finite nuclei turns out to be small.
There are three modifications of the fit protocol com-pared to [51, 52]. The obvious one is that the values for CJ
0 and C1J are fixed beforehand as the parameters
that will later on label and classify the fits. The second
is that we have added the binding energies of 90Zr and
-60 0 60 120 180 240 -60 0 60 120 180 240 10 20 30 40 χ2 (T11) β [MeV fm5] α [MeV fm5 ] χ2
FIG. 2: Values of the cost function χ2 as defined in the fit
procedure, for the set of parameterizations TIJ. The label “T11” indicates the position of this parameterization in the (α,β)-plane as obtained from Eqs. (36). Contour lines are
drawn at χ2 = 11, 12, 15, 20, 25, and 30. The minimum
value is found for T21 (χ2 = 10.05), the maximum for T61
(χ2= 37.11).
100Sn to the set of data. Indeed, we observed that the
latter nucleus is usually significantly overbound when not included in the fit. The third is that we have dropped
the constraint x2= −1 that was imposed on the SLy
pa-rameterizations [51, 52] to ensure the stability of infinite homogeneous neutron matter against a transition into a ferromagnetic state. On the one hand, this stability criterion is completely determined by the coupling con-stants of the time-odd terms in the energy functional [76], that we do not want to constrain here, accepting that the parameterizations might be of limited use beyond the present study. On the other hand, the tensor force brings many new contributions to the energy per parti-cle of polarized nuparti-clear matter that lead to a much more complex stability criterion. We postpone the entire dis-cussion concerning the stability in polarized systems in the presence of a tensor force to future work that will also address finite-size instabilities [84]. It also has to be stressed that the actual stability criterion, as all proper-ties of the time-odd part of the Skyrme energy functional, depends on the choices made for the interpretation of its coupling constants, i.e. antisymmetrized vertex or den-sity functional [76].
The properties of the finite nuclei entering the fit are computed using a Slater determinant without taking
pairing into account. The cost function χ2 was
parameteri--180 -150 -120 -90 -60 -30 0 30 60 90 120 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 B J [MeV fm1 5 ] BJ0 [MeV fm5] T11 T12 T13 T14 T15 T16 T21 T22 T23 T24 T25 T26 T31 T32 T33 T34 T35 T36 T41 T42 T43 T44 T45 T46 T51 T52 T53 T54 T55 T56 T61 T62 T63 T64 T65 T66
FIG. 3: The contributions from the tensor force BJ
0 and B1J
to the effective coupling constants of the J2term at sphericity.
Diagonal lines as in Fig. 1. The diagonal where BJ
0 + B1J =
αT = 0 (pure proton-neutron contribution) additionally
cor-responds to an isospin-singlet force with to≡U = 0.
zations can be found in the Physical Review archive [85].
Figure 2 displays the value of χ2after minimization as
a function of the recoupled coupling constants α and β. The first striking feature is the existence of a “valley” at
β = 0, i.e. a pure like-particle tensor term ∼ (J2
n+ J2p).
The abrupt rise of χ2around this value can be attributed
to the term depending on nuclear binding energies, as sharp variations of energy residuals can be seen between neighboring magic nuclei with functionals of the T6J
se-ries (β = 240). For example, 48Ca and 90Zr tend to be
significantly overbound in this case. We will come back later to discussing the implications for the quality of the functionals.
C. General properties of the fits
The coupling constants of the energy functional for spherical nuclei (28) obtained for T22 are very similar to those of SLy4, except for a slight readjustment coming
from the inclusion of the binding energies of 90Zr and
100Sn in the fit as well as the abandoned constraint on
x2. With its value of −0.945, the x2 obtained for T22
still stays close to the value −1 enforced for SLy4, which confirms that this is not too severe a constraint for
pa-rameterizations without effective J2 terms at sphericity.
Increasing the effective tensor term coupling constants CJ
t, however, the values for x2 start to deviate strongly
from the region around −1, which is to a large extent due
to the feedback from the contribution of the J2 terms to
-60 0 60 120 180 240 -60 0 60 120 180 240 100 150 200 W0 [MeV fm5] (T11) β [MeV fm 5 ] α [MeV fm5] W0 [MeV fm5]
FIG. 4: Value of spin-orbit coupling constant W0 for each
of the parameterizations TIJ, vs. indices I and J (The
“(T11)” label indicates the position of this parameterization
in the (α, β)-plane). The contour lines differ by 20 MeV fm5.
The values plotted here range from 103.7 MeV fm5 (T11) to
195.3 MeV fm5 (T66).
the surface and surface symmetry energy coefficients in the presence of constraints on isoscalar and isovector
ef-fective masses, all of which also depend on x2. A more
de-tailed discussion of the contribution of the J2terms to the surface energy coefficients will be given elsewhere [41].
From the constrained coupling constants CJ
0 and C1J,
the respective contributions BJ
0 and B1J from the tensor
force can be deduced afterwards using the expressions given in Sect. II C. Their values, shown in Fig. 3, are less regularly distributed, which is a consequence of the the non-linear interdependence of all coupling constants. Still, a general trend can be observed, such that all parameterizations are shifted towards the “south-west” compared to Fig. 1. In turn, this indicates that the con-tribution from the central Skyrme force always stays in
the small region outlined by SkP, SLy5, Zσ, etc in Fig. 1,
with values that range between 28 and 104 MeV fm5 for
AJ
0 and between 38 to 62 MeV fm5 for AJ1, respectively.
This justifies a posteriori to use the tensor force as a
motivation to decouple the J2
t terms from the central
part of the effective Skyrme vertex. We note in passing that all our parameterizations TI4 correspond to an al-most pure proton-neutron or isospin-singlet tensor force, i.e. the term ∝ tein Eq. (20), as they are all located close
to the αT = 0 line.
We also find a particularly strong and systematic
vari-ation of the coupling constant W0of the spin-orbit force,
which varies from W0 = 103.7 MeV fm5 for T11 to
W0= 195.3 MeV fm5 for T66, see Fig. 4. This variation
is of course correlated to the strength of the tensor force. As already shown, the tensor force has the tendency to reduce the spin-orbit splittings in spin-unsaturated nu-clei. To maintain a given spin-orbit splitting in such a
nucleus, the spin-orbit coupling constant W0 has to be
Ni, T44 0.015 0.010 0.005 0.000 ρsat/2 0 1 2 3 4 5 6 7 r [fm] 62 50 40 28 20 N -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 Jn [fm-4]
FIG. 5: (Color online) Radial component of the neutron
spin-orbit current for the chain of Ni isotopes, plotted against radius and neutron number N . The solid line on the base plot indicates the radius where the total density has half its saturation value.
IV. RESULTS AND DISCUSSION
The calculations presented below include open-shell nuclei treated in the Hartree-Fock-Bogoliubov (HFB) framework. In the particle-particle channel, we use a zero-range interaction with a mixed surface/volume form factor (called DFTM pairing in Ref. [86]). The HFB equations were regularized with a cutoff at 60 MeV in the quasiparticle equivalent spectrum [87]. The
pair-ing strength was adjusted in120Sn with the particle-hole
mean field calculated using the parameter set T33. The resulting strength was kept at the same value for all pa-rameterizations, which is justified by the fact that the effective mass parameters are the same. Moreover, we thus avoid including, in the adjustment of the pairing strength, local effects linked with changes in details of the single-particle spectrum.
A. Spin-orbit currents and potentials
As a first step in the analysis of the role of the tensor terms and their interplay with the spin-orbit interaction in spherical nuclei, we analyze the spin-orbit current den-sity and its relative contribution to the spin-orbit poten-tial. We choose the chain of nickel isotopes, Z = 28, as it covers the largest number of spherical neutron shells and subshells (N = 20, 28, 40 and 50) of any isotopic chain, two of which are spin-saturated (N = 20 and 40), while the other two are not. Figure 5 displays the radial
com-ponent of the neutron spin-orbit current Jnfor isotopes
from the proton to the neutron drip-lines. The calcula-tions are performed with T44, but the spin-orbit current is fairly independent from the parameterization. Starting
from N = 20, which corresponds to a completely filled and spin-saturated sd-shell, the next magic number at
N = 28 is reached by filling the 1f7/2 shell, which leads
to the steeply rising bump in the plot of Jn in the
fore-ground, peaked around r ≃ 3.5 fm. Then, from N = 28 to N = 40 the rest of the f p shell is filled, which first produces the small bump at small radii that corresponds
to the filling of the 2p3/2 shell, but ultimately leads to
a vanishing spin-orbit current when the 1f and 2p lev-els are completely filled for the N = 40 isostope, visible as the deep valley in Fig. 5. Adding more neutrons, the
filling of the 1g9/2 shell leads again to a strong neutron
spin-orbit current at N = 50. For the remaining isotopes
up to the neutron drip line, the evolution of Jnis slower
with the filling of the 2d and 3s orbitals.
A few further comments are in order. First, the spin-orbit current clearly reflects the spatial probability dis-tribution of the single-particle wave function in pairs of unsaturated spin-orbit partners. Within a given shell, the high-ℓ states contribute at the surface, represented by the solid line on the base of Fig. 5, while low-ℓ states contribute at the interior. The peak from the high-ℓ or-bitals, however, is always located on the inside of the nu-clear surface, as defined by the radius of half saturation density. Second, within a given shell, the largest contri-butions to the spin-orbit current density obviously come from the levels with largest ℓ, as they have the largest degeneracy factors in (27), and because they do not have nodes, which leads to a single, sharply peaked contribu-tion. Third, the spin-orbit current is not exactly zero for nominally “spin-saturated” nuclei, exemplified by the N = 20 and N = 40 isotopes in Fig. 5, as the radial single-particle wave functions are not exactly identical for all pairs of spin-orbit partners, which is a necessary
requirement to obtain Jn= 0 at all radii (Cf. the example
of the ν 2d states in132Sn in Fig. 16 below). Fourth, pair-ing and other correlations will always smooth the fluctu-ations of the spin-orbit current with nucleon numbers, as levels in the vicinity of the Fermi energy will never be completely filled or empty.
Next, we compare the contributions from the tensor terms and from the spin-orbit force to the spin-orbit po-tentials of protons and neutrons, Eq. (35). The contri-butions from the tensor force to the spin-orbit poten-tial are proportional to the spin-orbit currents of pro-tons and neutrons. For the Ni isotopes, the proton spin-orbit current is very similar to that of the neutrons at N = 28 displayed in Fig. 5. For the parameterization T44 we use here as an example, we have contributions from both proton and neutron spin-orbit currents, which come with equal weights. Their combined contribution
to the spin-orbit potential of the neutron Wn might be as
large as 4 MeV, see Fig. 6. This is more than a third of the maximum contribution from the spin-orbit force to
Wn, see Fig. 7. The latter is proportional to a
combina-tion of the gradients of the proton and neutron densities,
2∇ρn(r) + ∇ρp(r), see Eq. (35). As a consequence, it
Ni, T44 3 2 1 0 ρsat/2 0 1 2 3 4 5 6 7 r [fm] 62 50 40 28 20 N -1 0 1 2 3 4 Wn,t [MeV fm]
FIG. 6: (Color online) Contribution from the tensor terms to the neutron spin-orbit potential for the chain of Ni isotopes as obtained with the parameterization T44. The solid line on the base plot indicates the radius where the isoscalar density
ρ0 crosses half its saturation value.
Ni, T44 0 -2 -4 -6 -8 ρsat/2 0 1 2 3 4 5 6 7 r [fm] 62 50 40 28 20 N -10 -8 -6 -4 -2 0 2 Wn,so [MeV fm]
FIG. 7: (Color online) Contribution from the spin-orbit force to the neutron spin-orbit potential for the chain of Ni isotopes as obtained with the parameterization T44. The solid line on the base plot indicates the radius where the isoscalar density
ρ0 crosses half its saturation value.
with slowly and monotonically varying width, depth and position. Only limited local variations can be seen on the interior due to small variations of the density profile originating from the successive filling of different orbits. Furthermore, one can easily verify that the contribution from the spin-orbit force is peaked at the surface of the nucleus (the solid line on the base plot). The strongest variation of the depth of this potential occurs just be-fore the neutron drip line at N = 62, where is becomes wider and shallower due to the development of a diffuse
Ni, T44 0 -2 -4 -6 ρsat/2 0 1 2 3 4 5 6 7 r [fm] 62 50 40 28 20 N -8 -6 -4 -2 0 2 Wn [MeV fm]
FIG. 8: (Color online) Total neutron spin-orbit potential for the chain of Ni isotopes as obtained with the parameterization T44. The solid line on the base plot indicates the radius where
the isoscalar density ρ0crosses half its saturation value.
Ni, T44 0 -2 -4 -6 ρsat/2 0 1 2 3 4 5 6 7 r [fm] 62 50 40 28 20 N -8 -6 -4 -2 0 2 Wp [MeV fm]
FIG. 9: (Color online) Total proton spin-orbit potential for the chain of Ni isotopes as obtained with the parameterization T44. The solid line on the base plot indicates the radius where
the isoscalar density ρ0crosses half its saturation value.
neutron skin, which reduces the gradient of the neutron density [6, 7, 8].
Adding the contributions from the proton and neutron tensor terms to that from the spin-orbit force, the total neutron spin-orbit potential for neutrons in Ni isotopes is shown in Fig. 8. For the parameterization T44 used here (and most others in the sample of parameterizations used in this study) the dominating contributions from the spin-orbit and tensor forces to the spin-orbit
poten-tial are of opposite sign. For Ni isotopes, Jp is always
quite large, while Jn varies as shown in Fig. 5. Notably,
-25 -20 -15 -10 -5 0 20 30 40 50 60 εi [MeV] N (20) (28) (40) (50)
π
1g9/2 2p1/2 1f5/2 2p3/2 1f7/2 1d3/2 2s1/2 1d5/2 -25 -20 -15 -10 -5 0 εi [MeV] (20) (28) (40) (50)ν
2d3/2 3s1/2 2d5/2 1g9/2 2p1/2 1f5/2 2p3/2 1f7/2 2s1/2 1d3/2FIG. 10: (Color online) Single-particle spectra of neutrons
(upper panel) and protons (lower panel) for the chain of Ni isotopes, as obtained with the parameterization T22 with
van-ishing combined J2 terms. The thick solid line in the upper
panel denotes the Fermi energy for neutrons.
the combined contribution from the spin-orbit and tensor forces to the spin-orbit potential (35), one must keep in mind that they are peaked at different radii. Moreover, the variation of tensor-term coupling constants among a set of parameterizations implies a rearrangement of the spin-orbit term strength, as will be discussed later. As a consequence, taking into account the tensor force modi-fies the width and localization of the spin-orbit potential
Wq(r) much more than it modifies its depth through the
variation of the spin-orbit currents.
Our observations also confirm the finding of Otsuka
et al. [46] that the spin-orbit splittings might be more
strongly modified by the tensor force than they are by neutron skins in neutron-rich nuclei through the reduc-tion of the gradient of the density.
Figure 9 shows the spin-orbit potential of the protons for the chain of Ni isotopes. Here, the contribution from the spin-orbit force has a larger contribution coming from the gradient of the proton density that just grows with the mass number, without being subject to varying shell fluctuations. The same holds for the proton contribution from the tensor terms. Only the neutron contribution
from the tensor terms varies rapidly, proportional to Jn
displayed in Fig. 5, which has a very limited effect on the total spin-orbit potential, though.
With that, we can examine how the tensor terms af-fect the evolution of single-particle spectra. To that end, Fig. 10 shows the single-particle energies of protons and neutrons along the chain of Ni isotopes for the parameter-ization T22 with vanishing combined tensor terms, which
-25 -20 -15 -10 -5 0 20 30 40 50 60 εi [MeV] N (20) (28) (40) (50)
π
1g9/2 2p1/2 1f5/2 2p3/2 1f7/2 1d3/2 2s1/2 1d5/2 -25 -20 -15 -10 -5 0 εi [MeV] (20) (28) (40) (50)ν
2d3/2 3s1/2 2d5/2 1g9/2 2p1/2 1f5/2 2p3/2 1f7/2 2s1/2 1d3/2FIG. 11: (Color online) The same as Fig. 10, obtained with T44 with proton-neutron and like-particle tensor terms of equal strength.
will serve as a reference, while Fig. 11 shows the same for the parameterization T44 with proton-neutron and like-particle tensor terms of equal strength. For the latter, the variation of the neutron sporbit current with N in-fluences both neutron and proton single-particle spectra. The effect of the tensor terms is subtle, but clearly visi-ble: for T22, the major change of the single-particle en-ergies is their compression with increasing mass number, while for T44 the level distances oscillate on top of this background correlated to the neutron shell and sub-shell closures at N = 20, 28, 40 and 50. As shown above, the neutron spin-orbit current vanishes for N = 20, where it consequently has no effect on the spin-orbit potentials and splittings. By contrast, the neutron spin-orbit cur-rent is large for N = 28 and 50, where its contribution to the spin-orbit potential reduces the splittings from the spin-orbit force.
Pb, T44 0.010 0.005 0.000 ρsat/2 0 2 4 6 8 10 r [fm] 100120 140160 180 N -0.005 0.000 0.005 0.010 0.015 Jn [fm-4]
FIG. 12: (Color online) Radial component of the Neutron
spin-orbit current for the chain of Pb isotopes plotted in the same manner as in Fig. 5.
Note that both the orbit current J and the spin-orbit potential are exactly zero at r = 0 as they are vectors with negative parity.
B. Single-particle energies
As a next step, we analyze the modifications that the
presence of J2 terms brings to single-particle energies in
detail. Before we do so, a few general comments on the definition and interpretation of single-particle energies are in order. From an experimental point of view, empir-ical single-particle energies in a doubly-magic nucleus are determined as the separation energies between the even-even doubly magic nucleus and low-lying states in the adjacent odd-A nuclei, i.e. they are differences of bind-ing energies. In nuclear models, however, it is customary to discuss shell structure and single-particle energies in
terms of the spectrum of eigenvalues ǫi of the
Hartree-Fock mean-field Hamiltonian (in even-even nuclei), as we have done already in Figs. 10 and 11:
ˆ
h Φi = ǫiΦi. (37)
In the nuclear EDF approach without pairing, the ref-erence state is directly constructed as a Slater
determi-nant of eigenstates of ˆh; hence, the corresponding
eigen-values are directly connected to the fundamental build-ing blocks of the theory and reflect the mean field in the nucleus. The density of single-particle levels around the Fermi surface drives the magnitude of pairing cor-relations, the relative distance of single-particle levels at sphericity and their quantum numbers determine to a large extent the detailed structure of the deformation en-ergy landscape which in turn, determines the collective spectroscopy. The spectroscopic properties of even-even
nuclei, in particular when they exhibit shape coexistence, provide valuable benchmarks for the underlying single-particle spectrum [56]. The link between the spectrum of single-particle energies on the one hand and the col-lective excitation spectrum on the other hand, however, always remains indirect.
On the other hand, “single-particle” states near the Fermi level of a magic nucleus can be observed by adding or removing a particle in one of these states, and thus cor-respond to the ground and excited states of the neighbor-ing odd-mass nuclei. Assumneighbor-ing an infinitely stiff magic core, which is neither subject to any rearrangement or po-larization, nor to any collective excitations following the addition (or removal) of a nucleon, the separation ener-gies with the states in the odd-mass neighbors are equal to the single-particle energies as defined through (37). This highly idealized situation is modified by static [88] and dynamic [89, 90] correlations, often called “core po-larization” (see chapter 7 of Ref. [91]) and “particle-vibration coupling” (see section 9.3.3 of Ref. [92]) in the literature, that alter the separation energies. The main effect of the correlations is that they compress the spec-trum, pulling down the levels from above the Fermi en-ergy and pushing up those from below. The gross fea-tures, i.e. the ordering and relative placement of single-particle states, however, are more weakly affected by correlations. The particle-vibration coupling, however, is also responsible for the fractionization of the single-particle strength. When the latter is too large, the naive
comparison between the calculated ǫi given by Eq. (37)
and the energy of the lowest experimental state with the same quantum numbers is not even qualitatively mean-ingful anymore [48].
We mention that a part of the static correlations orig-inate from the non-vanishing time-odd densities in the mean-field ground-state of an odd-A nucleus, that also cannot be truly spherical, so that the complete energy functional from Eq. (23) should be considered in a fully self-consistent calculation of the separation energies.
The effective single-particle energies that are used to characterize the underlying shell structure in the inter-acting shell model [93] have a slightly different mean-ing. Their definition usually renormalizes polarization and particle-vibration coupling effects around a doubly-magic nucleus whereas their evolution is discussed in terms of monopole shifts [94]. A collection of effective single-particle energies and their evolution was collected by Grawe [95, 96]. Note that the SkX parameterization of the Skyrme energy functional by Brown and its vari-ants [48, 97] were constructed aiming at a description of effective single-particle energies along these lines.
It should be kept in mind that the obvious, coarse
dis-crepancies between the calculated spectra of ǫµ and the
